Engee documentation

DAE Solvers

For medium to low accuracy small numbers of DAEs in constant mass matrix form, the Rosenbrock23 and Rodas4 methods are good choices which will get good efficiency if the mass matrix is constant. Rosenbrock23 is better for low accuracy (error tolerance <1e-4) and Rodas4 is better for high accuracy. Another choice at high accuracy is Rodas5P and RadauIIA5.

Non-constant mass matrices are not directly supported: users are advised to transform their problem through substitution to a DAE with constant mass matrices.

If the problem cannot be defined in mass matrix form, the recommended method for performance is IDA from the Sundials.jl package if you are solving problems with Float64. If Julia types are required, currently DFBDF is the best method but still needs more optimizations.

Full List of Methods

Initialization Schemes

For all OrdinaryDiffEq.jl methods, an initialization scheme can be set with a common keyword argument initializealg. The choices are:

  • BrownFullBasicInit: For Index-1 DAEs implicit DAEs and semi-explicit DAEs in mass matrix form. Keeps the differential variables constant. Requires du0 when used on a DAEProblem.

  • ShampineCollocationInit: For Index-1 DAEs implicit DAEs and semi-explicit DAEs in mass matrix form. Changes both the differential and algebraic variables.

  • NoInit: Explicitly opts-out of DAE initialization.

OrdinaryDiffEq.jl (Implicit ODE)

These methods from OrdinaryDiffEq are for DAEProblem specifications.

  • DImplicitEuler - 1st order A-L and stiffly stable adaptive implicit Euler

  • DABDF2 - 2nd order A-L stable adaptive BDF method.

  • DFBDF - A fixed-leading coefficient adaptive-order adaptive-time BDF method, similar to ode15i or IDA in divided differences form.

OrdinaryDiffEq.jl (Mass Matrix)

These methods require the DAE to be an ODEProblem in mass matrix form. For extra options for the solvers, see the ODE solver page.

The standard Hermite interpolation used for ODE methods in OrdinaryDiffEq.jl is not applicable to the algebraic variables. Thus, for the following mass-matrix methods, use the interpolation (thus saveat) with caution if the default Hermite interpolation is used. All methods which mention a specialized interpolation (and implicit ODE methods) are safe.

Rosenbrock Methods

  • ROS3P - 3rd order A-stable and stiffly stable Rosenbrock method. Keeps high accuracy on discretizations of nonlinear parabolic PDEs.

  • Rodas3 - 3rd order A-stable and stiffly stable Rosenbrock method.

  • RosShamp4- An A-stable 4th order Rosenbrock method.

  • Veldd4 - A 4th order D-stable Rosenbrock method.

  • Velds4 - A 4th order A-stable Rosenbrock method.

  • GRK4T - An efficient 4th order Rosenbrock method.

  • GRK4A - An A-stable 4th order Rosenbrock method. Essentially "anti-L-stable" but efficient.

  • Ros4LStab - A 4th order L-stable Rosenbrock method.

  • Rodas4 - A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant

  • Rodas42 - A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant

  • Rodas4P - A 4th order A-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant. 4th order on linear parabolic problems and 3rd order accurate on nonlinear parabolic problems (as opposed to lower if not corrected).

  • Rodas4P2 - A 4th order L-stable stiffly stable Rosenbrock method with a stiff-aware 3rd order interpolant. 4th order on linear parabolic problems and 3rd order accurate on nonlinear parabolic problems. It is an improvement of Roadas4P and in case of inexact Jacobians a second order W method.

  • Rodas5 - A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant.

  • Rodas5P - A 5th order A-stable stiffly stable Rosenbrock method with a stiff-aware 4th order interpolant. Has improved stability in the adaptive time stepping embedding.

Rosenbrock-W Methods

  • Rosenbrock23 - An Order 2/3 L-Stable Rosenbrock-W method which is good for very stiff equations with oscillations at low tolerances. 2nd order stiff-aware interpolation.

  • Rosenbrock32 - An Order 3/2 A-Stable Rosenbrock-W method which is good for mildly stiff equations without oscillations at low tolerances. Note that this method is prone to instability in the presence of oscillations, so use with caution. 2nd order stiff-aware interpolation.

  • RosenbrockW6S4OS - A 4th order L-stable Rosenbrock-W method (fixed step only).

  • ROS34PW1a - A 4th order L-stable Rosenbrock-W method.

  • ROS34PW1b - A 4th order L-stable Rosenbrock-W method.

  • ROS34PW2 - A 4th order stiffy accurate Rosenbrock-W method for PDAEs.

  • ROS34PW3 - A 4th order strongly A-stable (Rinf~0.63) Rosenbrock-W method.

!!! note

Rosenbrock23 and Rosenbrock32 have a stiff-aware interpolation but this interpolation is not safe for the algebraic variables. Thus use the interpolation (thus saveat) with caution if the default Hermite interpolation is used.

FIRK Methods

  • RadauIIA5 - An A-B-L stable fully implicit Runge-Kutta method with internal tableau complex basis transform for efficiency.

SDIRK Methods

  • ImplicitEuler - Stage order 1. A-B-L-stable. Adaptive timestepping through a divided differences estimate via memory. Strong-stability preserving (SSP).

  • ImplicitMidpoint - Stage order 1. Symplectic. Good for when symplectic integration is required.

  • Trapezoid - A second order A-stable symmetric ESDIRK method. "Almost symplectic" without numerical dampening. Also known as Crank-Nicolson when applied to PDEs. Adaptive timestepping via divided differences on the memory. Good for highly stiff equations which are non-oscillatory.

Multistep Methods

Quasi-constant stepping is the time stepping strategy which matches the classic GEAR, LSODE, and ode15s integrators. The variable-coefficient methods match the ideas of the classic EPISODE integrator and early VODE designs. The Fixed Leading Coefficient (FLC) methods match the behavior of the classic VODE and Sundials CVODE integrator.

  • QNDF1 - An adaptive order 1 quasi-constant timestep L-stable numerical differentiation function (NDF) method. Optional parameter kappa defaults to Shampine’s accuracy-optimal -0.1850.

  • QBDF1 - An adaptive order 1 L-stable BDF method. This is equivalent to implicit Euler but using the BDF error estimator.

  • ABDF2 - An adaptive order 2 L-stable fixed leading coefficient multistep BDF method.

  • QNDF2 - An adaptive order 2 quasi-constant timestep L-stable numerical differentiation function (NDF) method.

  • QBDF2 - An adaptive order 2 L-stable BDF method using quasi-constant timesteps.

  • QNDF - An adaptive order quasi-constant timestep NDF method. Utilizes Shampine’s accuracy-optimal kappa values as defaults (has a keyword argument for a tuple of kappa coefficients).

  • QBDF - An adaptive order quasi-constant timestep BDF method.

  • FBDF - A fixed-leading coefficient adaptive-order adaptive-time BDF method, similar to ode15i or CVODE_BDF in divided differences form.

Sundials.jl

Note that this setup is not automatically included with DifferentialEquations.jl. To use the following algorithms, you must install and use Sundials.jl:

using Pkg
Pkg.add("Sundials")
using Sundials

  • IDA: A fixed-leading coefficient fully implicit BDF method. Efficient for large systems.

For more details on controlling the Sundials.jl solvers, see the Sundials detailed solver API page

DASKR.jl

DASKR.jl is not automatically included by DifferentialEquations.jl. To use this algorithm, you will need to install and use the package:

using Pkg
Pkg.add("DASKR")
using DASKR

  • daskr - This is a wrapper for the well-known DASKR algorithm.

For more details on controlling the DASKR.jl solvers, see the DASKR detailed solver API page

DASSL.jl

  • dassl - A native Julia implementation of the DASSL algorithm.

ODEInterfaceDiffEq.jl

These methods require the DAE to be an ODEProblem in mass matrix form. For extra options for the solvers, see the ODE solver page.

  • seulex - Extrapolation-algorithm based on the linear implicit Euler method.

  • radau - Implicit Runge-Kutta (Radau IIA) of variable order between 5 and 13.

  • radau5 - Implicit Runge-Kutta method (Radau IIA) of order 5.

  • rodas - Rosenbrock 4(3) method.